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36
History and evolution of the Density Theorem for Gabor frames
, 2007
"... The Density Theorem for Gabor Frames is one of the fundamental results of timefrequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arb ..."
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Cited by 40 (6 self)
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The Density Theorem for Gabor Frames is one of the fundamental results of timefrequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler–Raz biorthogonality relations, the Duality Principle, the Balian–Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.
Pseudodifferential operators on locally compact abelian groups and Sjöstrand’s symbol class
 J. Reine Angew. Math
, 2006
"... We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the KohnNirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard ana ..."
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Cited by 21 (2 self)
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We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the KohnNirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis ” techniques are not available on locally compact abelian groups, a new timefrequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjöstrand’s original results are thus understood as a phenomenon of abstract harmonic analysis rather than “hard analysis ” and are proved in their natural context and generality. 1
The noncommutative Wiener lemma, linear independence, and special properties of the algebra of timefrequency shift operators
 Trans. Amer. Math. Soc
"... In this paper we analyze the Banach *algebra of timefrequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators. As a corollary we ob ..."
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Cited by 14 (2 self)
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In this paper we analyze the Banach *algebra of timefrequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators. As a corollary we obtain a special case of the HeilRamanathanTopiwala conjecture regarding linear independence of finitely many timefrequency shifts of one L2 function. We also estimate the coefficient decay of the inverse of finite linear combinations of timefrequency shifts. 1
Microlocal analysis in Fourier Lebesgue and modulation spaces. Part II
, 2009
"... nsider different types of (local) products f1f2 in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wavefront properties. We also consider semilinear equations of the form P(x, D)f = G(x, Jkf), with appropriate polynomials ..."
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Cited by 13 (8 self)
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nsider different types of (local) products f1f2 in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wavefront properties. We also consider semilinear equations of the form P(x, D)f = G(x, Jkf), with appropriate polynomials P and G. If the solution locally belongs to appropriate weighted Fourier Lebesgue space FL q (ω)(Rd) and P is noncharacteristic at (x0, ξ0), then we prove that (x0, ξ0) ∈ (f), where ˜ω depends on ω, P and G.
Convergence analysis of the finite section method and Banach algebras of matrices
 OPER. THEORY
, 2008
"... The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted `pspaces. Our approach uses recent results f ..."
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Cited by 11 (0 self)
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The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted `pspaces. Our approach uses recent results from the theory of Banach algebras of matrices with offdiagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of unstructured nonhermitian matrices as well as to least squares problems.
Quantitative estimates for the finite section method
"... Abstract. The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. We present quantitative estimates for the rate of the convergence of the finite section method on weighted ℓ pspaces. Our approach uses recent results from the theory of ..."
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Cited by 10 (1 self)
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Abstract. The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. We present quantitative estimates for the rate of the convergence of the finite section method on weighted ℓ pspaces. Our approach uses recent results from the theory of Banach algebras of matrices with offdiagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of nonhermitian matrices. An example from digital communication illustrates the practical usefulness of the proposed theoretical framework.
A Noncommutative Wiener Lemma and a Faithful Tracial State ON BANACH ALGEBRAS OF TIMEFREQUENCY SHIFT OPERATORS
, 2005
"... In this paper we analyze the Banach *algebra of timefrequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which implies the algebra contains no compact operators. As a corollary we o ..."
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Cited by 9 (1 self)
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In this paper we analyze the Banach *algebra of timefrequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which implies the algebra contains no compact operators. As a corollary we obtain a special case of the HeilRamanathanTopiwala conjecture regarding linear independence of finitely many timefrequency shifts of one L² function.
InverseClosedness of a Banach Algebra of Integral Operators on the Heisenberg Group
"... Let H be the general, reduced Heisenberg group. Our main result establishes the inverseclosedness of a class of integral operators acting on L p (H), given by the offdiagonal decay of the kernel. As a consequence of this result, we show that if α1I +Sf, where Sf is the operator given by convolutio ..."
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Cited by 7 (0 self)
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Let H be the general, reduced Heisenberg group. Our main result establishes the inverseclosedness of a class of integral operators acting on L p (H), given by the offdiagonal decay of the kernel. As a consequence of this result, we show that if α1I +Sf, where Sf is the operator given by convolution with f, f ∈ L 1 v(H), is invertible in B(L p (H)), then (α1I + Sf) −1 = α2I + Sg, and g ∈ L 1 v(H). We prove analogous results for twisted convolution on a locally compact abelian group and its dual group. We apply the latter results to a class of Weyl pseudodifferential operators, and briefly discuss relevance to mobile communications. 1
Optimal adaptive computations in the Jaffard algebra and localized frames
 Bericht 20064, FB 12 Mathematik und Informatik, PhilippsUniversität Marburg
"... We study the efficient numerical solution of infinite matrix equations Au = f for a matrix A in the Jaffard algebra. These matrices appear naturally via frame discretizations in many applications such as Gabor analysis, sampling theory, and quasidiagonalization of pseudodifferential operators in t ..."
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Cited by 7 (1 self)
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We study the efficient numerical solution of infinite matrix equations Au = f for a matrix A in the Jaffard algebra. These matrices appear naturally via frame discretizations in many applications such as Gabor analysis, sampling theory, and quasidiagonalization of pseudodifferential operators in the weighted Sjöstrand class. The proposed algorithm has two main features: firstly, it converges to the solution with quasioptimal order and complexity with respect to classes of localized vectors; secondly, in addition to ℓ 2convergence, the algorithm converges automatically in some stronger norms of weighted ℓ pspaces. As an application we approximate the canonical dual frame of a localized frame and show that this approximation is again a frame, and even an atomic decomposition for a class of associated Banach spaces. The main tools are taken from adaptive algorithms, from the theory of localized frames, and the special Banach algebra properties of the Jaffard algebra.
Gabor representations of evolution operators
, 2013
"... We perform a timefrequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultraanalytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrödingertype propagators, ..."
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Cited by 7 (7 self)
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We perform a timefrequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultraanalytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrödingertype propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolictype operators. Besides the class of operators, the main novelty of the paper is the proof of superexponential (as opposite to superpolynomial) offdiagonal decay for the Gabor matrix representation.